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Universitat de Barcelona

Gran Via de Les Corts Catalanes 585 08007, Barcelona, Spain

e-mail: nunobarroso@ub.edu

Resumo: Discutimos a célebre demonstração do Último Teorema de Fermat

e as dificuldades que surgem ao tentar aplicar a mesma estratégia de prova sobre corpos de números. Terminamos com uma amostra dos resultados conhecidos no caso de corpos quadráticos.

Abstract We overview the celebrated proof of Fermat’s Last Theorem and

the challenges that arise when trying to carry it over to number fields. We conclude with a sample of the known results for quadratic fields.

palavras-chave: Fermat, modularidade, curvas elípticas. keywords: Fermat, modularity, elliptic curves.

1

Introduction

The search for a proof of Fermat’s Last Theorem (FLT) is one of the ri-chest and more romantic stories in the history of Mathematics. Remarkable progress in number theory as, for example, the origin of what is now alge-braic number theory and some incredible breakthroughs in the Langlands program, have come to light due to this pursuit.

Theorem 1 (FLT) The integer solutions to the Fermat equation

xn+ yn+ zn= 0 (1)

with n ≥ 3 are trivial, i.e., they satisfy xyz = 0.

The cases n = 3 and n = 4 of FLT were respectively solved by Euler and Fermat. From this it is easy to see that we only have to prove it for n = p ≥ 5 a prime. If (a, b, c) is a solution, then by scaling we can suppose that gcd(a, b, c) = 1; we call such a solution primitive. The Fermat equation, viewed as defining a curve in P2, has genus (p − 1)(p − 2)/2, and a celebrated theorem of Faltings tells us that there are only finitely many primitive solutions to (1), for each fixed n = p. Despite the efforts of many

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great mathematicians through 350 years, it was only in 1995 that a complete proof was published. In this paper, we will discuss this modern approach to FLT due to Hellegouarch, Frey, Serre and Ribet which culminated in Wiles’ proof [24] and created a new way of tackling Diophantine equations known as the modular method.

2

The modular method

The proof of FLT is based on three main pillars: Mazur’s irreducibility theorem, Wiles’ modularity theorem for semistable elliptic curves over Q and Ribet’s level lowering theorem. Explaining these pillars will involve a detour into some of the most fascinating areas of modern number theory: elliptic curves, Galois representations, modular forms and modularity. For a comprehensive introduction to these topics we suggest [3, 21, 22]. For an overview and history of various methods to study the Generalized Fermat equation xr+ yq= zp we refer to [1], which we follow closely in this section.

2.1 Elliptic curves

Let K be a field. The simplest definition of an elliptic curve E over K is: a smooth curve in P2 given by an equation of the form

E : y2z + a1xyz + a3yz2= x3+ a2x2z + a4xz2+ a6z3, (2)

with a1, a2, a3, a4 and a6∈ K. If the characteristic of K is not 2 or 3, then we can transform to a much simpler model given by the affine equation

E : Y2 = X3+ aX + b, (3)

where a and b ∈ K, whose disriminant is

E = −16(4a3+ 27b2).

We call (3) a Weierstrass model of E with discriminant ∆E. The

require-ment that E is smooth is equivalent to the assumption that ∆E 6= 0. There is another very important quantity attached to an elliptic curve called the j-invariant which can be computed from the model (3) by the formula

jE =

(−48a)3

E

.

Note however that jE is an invariant of the isomorphism class of E over K, the algebraic closure of K, and so independent of the choosen model.

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There is a distinguished K-point, the ‘point at infinity’, which we denote by ∞. Given a field L ⊇ K, the set of L-points on E is given by

E(L) = {(x, y) ∈ L2 : y2 = x3+ ax + b} ∪ {∞}.

It turns out that the set E(L) has the structure of an abelian group with ∞ as the identity element. The group structure is easy to describe geometrically: three points P1, P2, P3 ∈ E(L) add up to the identity element if and only if

there is a line ` defined over L meeting E in P1, P2, P3 (with multiplicities counted appropriately). The classic Mordell–Weil Theorem states that for a number field K the group E(K) is finitely generated. For a model as in (3), the 2-torsion subgroup E[2] consists of the points with y = 0 plus ∞. It turns out that the proofs by Euler and Fermat of FLT for n = 3, 4 are simply special cases of what are now standard Mordell–Weil group computations, as discussed in [1, Examples 1 and 2].

2.2 Modular forms

Let H = {z ∈ C : Im(z) > 0}. Let k and N be positive integers and set Γ0(N ) = ( a b c d ! ∈ SL2(Z) : c ≡ 0 (mod N ) ) ,

which is a subgroup of SL2(Z) of finite index. The group Γ0(N ) acts on H via fractional linear transformations

a b c d

!

: H → H, z 7→ az + b cz + d.

The quotient Y0(N ) = Γ0(N )\H has the structure of a non-compact Rie-mann surface. This has a standard compactification denoted X0(N ) and the difference X0(N ) \ Y0(N ) is a finite set of points called the cusps.

A modular form f of weight k and level N is a function f : H → C that satisfies the following conditions:

(i) f is holomorphic on H; (ii) f satisfies the property f

az + b cz + d



= (cz + d)kf (z), (4) for all z ∈ H and (a bc d) ∈ Γ0(N );

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(iii) f extends to a function that is holomorphic at the cusps.

It follows from these properties, and the fact that one of the cusps is the cusp at i∞, that f must have a Fourier expansion

f (z) =X

n≥0

cnqn where q(z) = exp(2πiz). (5)

It turns out that the set of modular forms of weight k and level N , denoted by Mk(N ), is a finite-dimensional vector space over C. A cusp form of weight k and level N is an f ∈ Mk(N ) that vanishes at all the cusps. As q(i∞) = 0 we see in particular that a cusp form must satisfy c0 = 0. The

cusp forms naturally form a subspace of Mk(N ) which we denote by Sk(N ). There is a natural family of commuting operators Tn : S2(N ) → S2(N ) (with n ≥ 1) called the Hecke operators. The eigenforms of level N are the weight 2 cusp forms that are simultaneous eigenvectors for all the Hecke operators. Such an eigenform is called normalized if c1 = 1 and thus its Fourier expansion has the form

f = q +X

n≥1

cnqn.

2.3 Modularity

Let E/Q be given by a model (2) where the ai ∈ Z, and having

(non-zero) discriminant ∆E ∈ Z. Carrying out a suitable linear substitution, we generally work with a minimal model: that is one where the ai ∈ Z and with

discriminant having the smallest possible absolute value. Associated to E is another, more subtle, invariant called the conductor NE, which we shall not define precisely, but we merely point that it is a positive integer sharing the same prime divisors as the minimal discriminant; that it measures the ‘bad behavior’ of the elliptic curve E modulo primes; and that it can be computed easily through Tate’s algorithm [22, Chapter IV]. In particular, the primes p not dividing NE are the primes of good reduction while those

satisfying pkNE are the primes of multiplicative reduction.

Now let p - ∆E be a prime. Reducing modulo p a minimal equation (2)

we obtain an elliptic curve ˜E over Fp. The set ˜E(Fp) is an abelian group as

before, but now necessarily finite, and we denote its order by # ˜E(Fp). Let ap(E) = p + 1 − # ˜E(Fp).

We are now ready to state a version of the modularity theorem due to Wiles, Breuil, Conrad, Diamond and Taylor [2, 23, 24]. This remarkable theorem was previously known as the Shimura–Taniyama conjecture.

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Theorem 2 (The Modularity Theorem) Let E/Q be an elliptic curve

with conductor NE. There exists a normalized eigenform f = q +P

cnqn of

weight 2 and level NE with cn∈ Z for all n, and such that for every prime

p - ∆E we have cp = ap(E).

For an elliptic curve E and an eigenform f as in this theorem we will also say that f corresponds to E via modularity.

2.4 Galois representations

Let E be an elliptic curve over C. The structure of the abelian group E(C) is particularly easy to describe. There is a discrete lattice Λ ⊂ C of rank 2 (that is, as an abelian group Λ ' Z2) depending on E, and an isomorphism

E(C) ' C/Λ. (6)

Let p be a prime. By the p-torsion of E(C) we mean the subgroup E[p] = {Q ∈ E(C) : pQ = 0}.

It follows from (6) that

E[p] ' (Z/pZ)2, (7)

which can be viewed as 2-dimensional Fp-vector space. Now let E be an

elliptic curve over Q. Then we may view E as an elliptic curve over C, and with the above definitions obtain an isomorphism E[p] ' (Z/pZ)2. However, in this setting, the points of E[p] have algebraic coordinates, and are acted on by GQ := Gal(Q/Q), the absolute Galois group of the rational numbers. Via the isomorphism (7), the group GQ acts on (Z/pZ)2. Thus we obtain a 2-dimensional representation depending on E/Q and the prime p:

ρE,p : GQ → GL2(Fp). (8)

We say that theρE,p is reducible if the matrices of the image ρE,p(GQ) share some common eigenvector. Otherwise we say that ρE,p is irreducible. We have now given enough definitions to be able to state Mazur’s theorem; this is often considered as the first step in the proof of FLT.

Theorem 3 (Mazur [15]) Let E/Q be an elliptic and p a prime.

(i) If p > 163, then ρE,p is irreducible.

(ii) If E has full 2-torsion (that is E[2] ⊆ E(Q)), square-free conductor and p ≥ 5, then ρE,p is irreducible.

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2.5 Ribet’s level lowering theorem

Let E/Q be an elliptic curve and associated mod p Galois representation ρE,p : GQ → GL2(Fp) as above. Let f be an eigenform. Deligne and Serre showed that such an f gives rise, for each prime p, to a Galois representation ρf,p : GQ → GL2(Fpr), where r ≥ 1 depends on f . If E corresponds to f

via the Modularity Theorem, then ρE,p ∼ ρf,p (the two representations are isomorphic). Thus the representation ρE,p is modular in the sense that it arises from a modular eigenform. Recall also from the Modularity Theorem that, if f corresponds to E via modularity, then the conductor of E is equal to the level of f . Sometimes it is possible to replace f by another eigenform of smaller level which has the same mod p representation. This process is called level lowering. We now state a special case of Ribet’s level lowering theorem. For a prime `, we let υ`(x) denote the `-adic valuation of x ∈ Q.

Theorem 4 (Ribet’s level lowering theorem [18]) Let E/Q be an

el-liptic curve with minimal discriminant ∆ and conductor N . Let p ≥ 3 be prime. Suppose that (i) the curve E is modular and (ii) the mod p repre-sentation ρE,p is irreducible. Let

Np = N Mp , where Mp= Y `||N, p | υ`(∆) `. (9)

Then ρE,p∼ ρg,p for some eigenform g of weight 2 and level Np.

We now know, by the Modularity Theorem that all elliptic curves over Q are modular, so condition (i) in Ribet’s theorem is automatically satisfied. We include it here both for historical interest but also because analogous level lowering results are available over other fields and modularity of all elliptic curves is still an open question over general fields.

2.6 The proof of Fermat’s Last Theorem

Suppose p ≥ 5 is prime, and a, b and c are non-zero pairwise coprime integers satisfying (1) with n = p. We reorder (a, b, c) so that

b ≡ 0 (mod 2) and ap ≡ −1 (mod 4). (10) We consider the Frey–Hellegouarch curve which depends on (a, b, c):

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whose minimal discriminant and conductor are: ∆ = a 2pb2pc2p 28 , N = Y `|∆ `.

Note that the conductor is square-free; conditions (10) ensure that 2 || N . The 2-torsion subgroup of E is E[2] = {∞, (0, 0), (ap, 0), (−bp, 0)} ⊂ E(Q). As p ≥ 5, we know by part (ii) of Mazur’s irreducibility theorem that ρE,p is irreducible. Moreover, E is modular by the Modularity Theorem1, and so the hypotheses of Ribet’s theorem are satisfied. We compute Np = 2 using

the recipe in (9). It follows thatρE,p∼ ρg,p, where g has weight 2 and level 2. But there are no eigenforms of weight 2 and level 2, a contradiction.

2.6.1 Some Historical Remarks

In the early 1970s, Hellegouarch had the idea of associating to a non-trivial solution of the Fermat equation the elliptic curve (11); he noted that the number field generated by its p-torsion subgroup E[p] has surprisingly little ramification. In the early 1980s, Frey observed that this elliptic curve enjoys certain remarkable properties that should rule out its modularity. Motivated by this, in 1985 Serre made precise his modularity conjecture and showed that it implies Fermat’s Last Theorem. Serre’s remarkable paper [20] also uses several variants of the Frey–Hellegouarch curve to link modularity to other Diophantine problems. Ribet announced his level-lowering theorem 1987, showing that modularity of the Frey–Hellegouarch curve implies FLT.

3

Fermat’s Last Theorem over Number Fields

3.1 Historical background

Interest in the Fermat equation over various number fields goes back to the 19th and early 20th Century. For example, Dickson’s History of the Theory of Numbers [4, pages 758 and 768] mentions extensions by Maillet (1897) and Furtwängler (1910) of classical ideas of Kummer to the Fermat equation xp + yp = zp (p > 3 prime) over the cyclotomic field Q(ζp). However, the elementary, cyclotomic and Mordell–Weil approaches to the Fermat equa-tion have had limited success. Indeed, even over Q, no combinaequa-tion of these

1

In fact, we only need modularity of semistable elliptic curves over Q, i.e. those with square-free conductor, which was the original modularity result proved by Wiles.

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approaches is known to yield a proof of FLT for infinitely many prime ex-ponents p. It is therefore natural to attempt to carry Wiles’ proof over to general number fields. The first work in this direction is due to Jarvis and Meekin [12] who showed FLT holds for the field Q(√2). They further analyzed the situation over other real quadratic fields to conclude that

“. . . the numerology required to generalise the work of Ribet and Wiles directly continues to hold for Q(2). . . there are no other real quadratic fields for which this is true . . . ”

3.2 The asymptotic Fermat’s conjecture

Let K be a number field and OK its ring of integers. By the Fermat equation with exponent p over K we mean

xp+ yp+ zp = 0, x, y, z ∈ OK. (12)

A solution (a, b, c) of (12) is called trivial if abc = 0, otherwise non-trivial. Clearly, over any K there are trivial solutions, such as (1, −1, 0), but some-times more, for example,

(18 + 17 √ 2)3+ (18 − 17 √ 2)3= 423, (1 +√−7)4+ (1 −−7)4 = 24,

showing that the exact same statement as of FLT does not hold over Q(√2) or Q(√−7). Instead it makes sense to consider the question only for large enough exponents. More precisely, we will say that the asymptotic

Fer-mat’s Last Theorem over K holds if there is some bound BK such that

for prime p > BK, all solutions to the Fermat equation (12) are trivial.

Now let K = Q(−3) and consider the element ω =

√ −3−1

2 . We have

that ω3= 1 and it is easy to see that, for all primes p ≥ 5, the equality ωp+ (ω2)p+ 1p= 0,

holds, hence the asymptotic FLT does not hold over Q(√−3).

Conjecture 1 (Asymptotic Fermat’s Conjecture) Let K be a number

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4

The modular method over totally real fields

We restrict ourselves to totally real fields, i.e., number fields such that all embeddings into C have image in R. This is a natural restriction, because modularity related objects and questions are very poorly understood for fields with at least one complex embedding, consequently all results about FLT for such fields are conditional on two deep conjectures of the Langlands program (see [19] for details). In contrast, for a totally real field K there is a well established theory of Hilbert modular forms which are the natural replacement for the modular forms over Q; it is not our objective to discuss details of this theory here. The only thing to keep in mind is that they satisfy the analogous properties over K to those described in §2.2 and that modularity of elliptic curves over K can be defined by a correspondence with Hilbert eigenforms, similar to the discussion in §2.3

In particular, since K is totally real, we have ω 6∈ K and we expect the Asymptotic Fermat Conjecture to hold for K. To properly discuss the challenges we face it helps to break the method into the following steps:

1. Constructing a Frey curve. Attach a Frey elliptic curve E/K to a

putative solution of (12).

2. Modularity. Prove modularity of E/K.

3. Irreducibility. Prove irreducibility of ρE,p, the mod p Galois repre-sentation attached to E.

4. Level lowering. Conclude that ρE,p ∼ ¯ρf,p where f is a Hilbert

ei-genform over K of (parallel) weight 2 and level among finitely many possibilities Ni. Here, ¯ρf,p denotes the mod p Galois representation

attached to f for some p | p in the field of coefficients Qf of f.

5. Contradiction. Compute all the eigenforms f predicted in Step 4 and

show thatρE,p 6∼ ¯ρf,p for all of them.

Suppose that a, b, c ∈ OK is a solution to (12) such that abc 6= 0. Since (12)

is equation (1) over K we consider in step 1 the classical Frey curve over K: Ea,b,c : Y2 = X(X − ap)(X + bp). (13)

4.1 The case of Q(√2)

Recall that steps 2–4 in the proof of FLT over Q are covered respectively by the three remarkable theorems of Wiles, Mazur and Ribet. However, at the

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time of writing [12] only step 4 was known to hold over a general K (due to the combined work of Jarvis, Rajae and Fijuwara [9, 10, 17]). To com-plete the modularity step Jarvis and Meekin showed that, under some non-restrictive assumptions on a, b, c (analogous to (10)), the curve Ea,b,cis

semis-table and its modularity followed by a result of Jarvis–Manoharmayum [11] stating that all semistable elliptic curves over Q(√2) are modular. For the irreducibility part they applied a criterion of Kraus [14] for p ≥ 17 and, fi-nally, a contradiction follows because after completing step 4 there are again no eigenforms.

4.2 The contradiction step

By looking at the proofs of FLT over Q and Q(√2), the reader may wonder why is there a step 5, as the contradiction happens automatically. It turns out these are the only cases where such convenient coincidence occurs. For example, if the class number of K is > 1, we cannot assume coprimality of a, b, c, and the Frey curve will not be semistable, consequently the levels obtained after level lowering will have larger norms and, in general, there are eigenforms at these levels. In fact, step 5 is nowadays the most difficult part when applying the modular method to solve (12) or any other Diophantine equation assuming, of course, that an associated Frey curve exists (i.e. step 1 can be done); unfortunately, there are only a few Diophantine equations known to have attached Frey curves.

4.3 Modularity and Irreducibility

Although the modularity of the Frey curve was the hardest step in the proof of FLT, nowadays we know it holds in full generality due to a result of Freitas–Le Hung–Siksek [6].

Theorem 5 (F.–Le Hung–Siksek) Let K be a totally real field. Up to

isomorphism over K, there are at most finitely many non-modular elliptic curves E over K.

Moreover, if K is quadratic, then all elliptic curves over K are modular. Furthermore, in the recent work of Derickx–Najman–Siksek [5], modu-larity of elliptic curves was extended to the case of totally real cubic fields.

Theorem 6 (Derickx–Najman–Siksek) All elliptic curves over totally

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These modularity results have the following important consequence.

Corollary 4.1 Let K be a totally real field. There is some constant AK,

depending only on K, such that for any non-trivial solution (a, b, c) of the Fermat equation (12) with prime exponent p > AK, the Frey curve Ea,b,c

given by (13) is modular.

Moreover, if K is quadratic or cubic then AK = 0.

There is no irreducibility result for ¯ρE,p over K analogous to Mazur’s

theorem. Instead, we can derive the following result from the works of David and Momose, who build on Merel’s Uniform Boundedness Theorem [16].

Theorem 7 Let K be a totally real field. There is a constant CK, depending

only on K, such that the following holds. If p > CK is prime, and E is an elliptic curve over K with either good or multiplicative reduction at all q | p, then ρE,p is irreducible.

4.4 A refined level lowering

The next step in the strategy is level lowering which is known to hold for general K due to the combined work of Fujiwara, Jarvis and Rajaei. As explained above, after applying level lowering, we will not obtain a contra-diction due to the presence of eigenforms. Instead, the idea is to use finer properties of the Frey curve Ea,b,c, to show that many of the eigenforms are

not a real obstruction.

Before proceeding, it is helpful here to make a comparison with the equation xp+ yp+ Lαzp= 0 over Q, with L an odd prime and α a positive

integer, considered by Serre and Mazur [20, p. 204]. A non-trivial solution to this latter equation gives rise, via modularity and level lowering, to a classical weight 2 newform f of level 2L; for L ≥ 13 there are such eigenforms and we face the same difficulty. Mazur however shows that if p is sufficiently large then f corresponds to an elliptic curve E0 with full 2-torsion and conductor 2L, and by classifying such elliptic curves concludes that L is either a Fermat or a Mersenne prime. To be able to transfer and refine Mazur’s argument to our setting, we need the following conjecture, which is the opposite direction to modularity and generalizes the Eichler–Shimura Theorem over Q.

Conjecture 2 (“Eichler–Shimura”) Let K be a totally real field. Let f

be a Hilbert newform of level N and parallel weight 2, and rational field of coefficients. Then there is an elliptic curve Ef/K with conductor N having the same L-function as f.

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We will also need some more notation. For K a totally real field, an element x ∈ K and a prime ideal q in OK we write υq(x) to denote a q-adic valuation

of x. Moreover, let

S = {P : P is a prime ideal of OK dividing 2OK},

T = {P ∈ S : OK/P = F2}, U = {P ∈ S : 3 - υP(2)}.

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We choose a set H of prime ideals m 6∈ S representing the elements in the class group of K. We also need an assumption, which we refer to as (ES):

(ES)      either [K : Q] is odd; or T 6= ∅;

or Conjecture 2 holds for K.

For a non-trivial solution (a, b, c) to the Fermat equation (12), let

Ga,b,c:= aOK+ bOK+ cOK. (15)

Now Mazur’s argument adapted to our setting gives the following result.

Theorem 8 Let K be a totally real field satisfying (ES). There is a

cons-tant BK, depending only on K, such that the following holds. Let (a, b, c) be a non-trivial solution to (12) with prime exponent p > BK, and rescale

(a, b, c) so that it remains integral and satisfies Ga,b,c= m for some m ∈ H. Write E for the Frey curve (13). Then there is an elliptic curve E0 over K such that

(i) the conductor of E0 is divisible only by primes in S ∪ {m}; (ii) #E0(K)[2] = 4;

(iii) ρE,p∼ ρE0,p;

Write j0 for the j-invariant of E0. Then, (a) for P ∈ T , we have υP(j0) < 0;

(b) for P ∈ U , we have either υP(j0) < 0 or 3 - υP(j0); (c) for q /∈ S, we have υq(j0) ≥ 0.

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5

Results over totally real fields

5.1 S-unit equations

Assuming (ES), Theorem 8 implies that a non-trivial solution to the Fermat equation over K with sufficiently large exponent p yields an elliptic curve E0/K with full 2-torsion and potentially good reduction away from the set S of primes above 2. There are such elliptic curves over every K, for example the curve Y2 = X3− X, and so we still do not get a simple contradiction. Note however that the latter elliptic curve does not satisfy conclusion (a) of Theorem 8 when the field K is such that T 6= ∅, hence it is not an obstruction in that case. An element x ∈ K is called an S-unit if υq(x) = 0 for all q 6∈ S. Using the fact that elliptic curves with full 2-torsion and good reduction away from S are classified by solutions to S-unit equations, we have the following result that describes when there are no E0/K as in Theorem 8, and so no obstruction to the desired contradiction.

Theorem 9 (F.–Siksek) Let K be a totally real field satisfying (ES). Let

S, T and U be as in (14). Write OSfor the group of S-units of K. Suppose that for every solution (λ, µ) to the S-unit equation

λ + µ = 1, λ, µ ∈ OS∗ (16)

there is

(A) either some P ∈ T that satisfies max{|υP(λ)|, |υP(µ)|} ≤ 4 υP(2),

(B) or some P ∈ U that satisfies both max{|υP(λ)|, |υP(µ)|} ≤ 4 υP(2), and υP(λµ) ≡ υP(2) (mod 3).

Then the asymptotic Fermat’s Last Theorem holds over K.

For all fields K, equation (16) has solutions in Q ∩ OS, namely (λ, µ) =

(2, −1), (−1, 2), (1/2, 1/2) which correspond to the elliptic curve Y2= X3− X, however these solutions satisfy (A) if T 6= ∅ and (B) if U 6= ∅.

5.2 The quadratic case

In view of Theorem 9, we have to solve the S-unit equation (16) and test the solutions in order to decide whether the asymptotic FLT holds over K. There are algorithms that, in principle, could do that for each particular K, but what is more interesting is to show that asymptotic FLT holds for infinite

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families of fields. For this we need to control the solutions to (16) over varying fields. This can be very hard but, in the case of real quadratic fields, we achieved considerable success, as illustrated by the following series of theorems taken from the joint works with Siksek [7, 8].

Theorem 10 Let d ≥ 2 be square-free, such that d ≡ 6, 10 (mod 16) or

d ≡ 3 (mod 8). Then the asymptotic FLT holds over Q(d).

Moreover, for d > 5 satisfying d ≡ 5 (mod 8), the same is true assuming that Conjecture 2 holds over Q(d).

There are other explicit congruence conditions on d for which asymptotic FLT is known to hold over Q(√d) (see [7, Theorem 1]) and, moreover, there are also real quadratic fields Q(√d) not given by a congruence condition on d for which asymptotic FLT holds. We have the following density theorem.

Theorem 11 (F.–Siksek) The asymptotic FLT holds for a set of real

qua-dratic fields of density 5/6. Assuming Conjecture 2 this density becomes 1. The following result shows that it is possible to optimize BK by

ma-king K concrete. In this case the proof does not pass through Theorem 9, but instead one needs to optimize the exponent bound at every step of the strategy, which raises other challenges not discussed here.

Theorem 12 (F.–Siksek) Let 3 ≤ d ≤ 23 be square-free and d 6= 5, 17 or

d = 79. Then, all solutions (a, b, c) to the equation xp+ yp+ zp = 0, a, b, c ∈ Q(

d) p ≥ 5 prime

satisfy abc = 0. Moreover, the same is true over Q(17) for half the expo-nents, more precisely, for all primes p ≥ 5 such that p ≡ 3, 5 (mod 8).

5.2.1 FLT over Q(√5)

Note that the field Q(√5) is not covered by any of the theorems above and indeed asymptotic FLT over Q(√5) seems to be a very hard open problem. The main reason being that the modular method does not see the difference between the Fermat equation (12) and its variant with unit coefficients

1 +√5 2 x p+1 − √ 5 2 y p+ zp= 0

which has a solution (1, 1, −1). Nevertheless, the following result gives evi-dence that FLT should be true over Q(√5), as predicted by Conjecture 1.

Theorem 13 (Kraus [13]) Let K = Q(5) and p < 107 be a prime. Then the Fermat equation with exponent p over K has only the trivial solutions.

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References

[1] M. A. Bennett, P. Mihăilescu and S. Siksek The Generalized Fermat Equation, pages 173–205 of Open Problems in Mathematics (J. F. Nash, Jr. and M. Th. Rassias eds), Springer, New York, 2016.

[2] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), 843–939.

[3] F. Diamond and J. Shurman, A First Course on Modular Forms, GTM

228, Springer, 2005.

[4] L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, New York, 1971.

[5] M. Derickx, F. Najman and S. Siksek, Elliptic curves over real cubic fields are modular, preprint.

[6] N. Freitas, B. V. Le Hung and S. Siksek, Elliptic curves over real quadra-tic fields are modular, Inventiones Mathemaquadra-ticae 201 (2015), 159–206. [7] N. Freitas and S. Siksek, The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields, Compos. Math. 151 (2015), no. 8, 1395–1415.

[8] N. Freitas and S. Siksek, Fermat’s Last Theorem over some small real quadratic fields, Algebra & Number Theory 9 (2015), no. 4, 875–895. [9] K. Fujiwara, Level optimisation in the totally real case,

arXiv:0602586v1, 27 February 2006.

[10] F. Jarvis Correspondences on Shimura curves and Mazur’s principle at p, Pacific J. Math., 213 (2), 2004, 267–280.

[11] F. Jarvis and J. Manoharmayum, On the modularity of supersingular elliptic curves over certain totally real number fields, Journal of Number Theory 128 (2008), no. 3, 589–618.

[12] F. Jarvis and P. Meekin, The Fermat equation over Q(√2), J. Number Theory 109 (2004), 182–196.

[13] A. Kraus, Sur le théorème de Fermat sur Q(√5), Annales Mathémati-ques du Québec 39.1 (2015), 49–59.

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[14] A. Kraus, Courbes elliptiques semi-stables et corps quadratiques, J. of number theory 60 (1996), 245–253

[15] B. Mazur, Rational isogenies of prime degree, Inventiones Math. 44 (1978), 129–162.

[16] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449.

[17] A. Rajaei, On the levels of mod ` Hilbert modular forms, J. reine angew. Math. 537 (2001), 33–65.

[18] K. A. Ribet, On modular representations of Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990), 431–476.

[19] M. H. Şengün and S. Siksek, On the asymptotic Fermat’s Last Theo-rem over numbers fields, Commentarii Mathematici Helvetici 93 (2018), 359–375.

[20] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Q/Q), Duke Math. J. 54 (1987), 179–230.

[21] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Sprin-ger, 1986.

[22] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer, 1994.

[23] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke al-gebras, Ann. of Math. 141 (1995), 553–572.

[24] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141 (1995), 443–551.

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